Numerical Results for the PCSS by New Dynamic Surface Control

Since the partner A, new Duffing-Van der Pol system, is described as

( )

where a, b, c, d, f are uncertain parameters. The partner B is described as

( )

The chaotic Lü system is chosen as functional system [45] and the augmented state

variable is z₄ =z₁²:

Now n=4. By NDSC, the error dynamics Eq. (4.15) becomes:

( )

and the boundary layer error dynamics Eq. (4.13) becomes:

( )

select controllers, estimated parameter dynamics, and the m as:

The time derivative of V is

2 2 2 2 2 2 2 2

1 2 3 4 1 2 3 4 0

V = − −e e −e −e −s −s −s −s ≤ (4.29) which is negative semi-definite function for e₁, e₂, e₃, e₄, s₁, s₂, s₃, s₄, a ,

, , ,

b c d f . The Lyapunov asymptotical stability theorem cannot be satisfied in this case. The common origin of error dynamics, parameter update dynamics, and boundary layer error dynamics cannot be concluded to be asymptotically stable. By pragmatical asymptotical theorem, D is a 13-manifold, n = 13 and the number of error state variables p = 8. When e = ₁ e = ₂ e = ₃ e = ₄ s = ₁ s = ₂ s = ₃ s = 0 and ⁴

a

, b , c , d , f take arbitrary values , V =0, so X is a 5-manifold, m = n - p = 13 – 8

= 5. m + 1 < n are satisfied. By the pragmatical asymptotical stability theorem, the common origin of error dynamics (4.23), boundary layer error dynamics (4.24), and parameter dynamics (4.27) are asymptotically stable. The equilibrium point e = ₁ e ₂

= e = ₃ e = ₄ s = ₁ s = ₂ s = ³ s = a = b = c = d = ⁴ f = 0 is pragmatically asymptotically stable. The PCSS is achieved under this scheme.

In this numerical simulation, we select the “unknown” parameter and initial states of the partner A and of functional system as a=0.01, b=1, c=5, d=0.67, f=0.05, g=36, h=20, k=3 to ensure the chaotic behavior. The initial states of those system are

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 0 2, 2 0 2.4, 3 0 5, 4 0 6, 1 0 5, 2 0 5, 3 0 10,

x = x = x = x = y = y = y =

( ) ( ) ( ) ( ) ( )

4 0 10, 1 0 2 0 3 0 4 0 10

y = z =z =z =z = . The estimated parameters have initial conditions ^aˆ

( )

^{0 =bˆ}

( ) ( )

^{0 =cˆ 0 =dˆ}

( )

^{0 = fˆ}

( )

^{0 =0}. The numerical results are shown in Fig. 4.1 ~ Fig. 4.4.

Fig. 4.1 Time histories of errors.

Fig. 4.2 Time histories of the differences of uncertain parameters and estimated parameters.

Fig. 4.3 Time histories of boundary layer errors.

Fig. 4.4 Time histories of m which is a bounded function of time and approaches to zero.

Chapter 5

Chaos Generalized Synchronization of New

Duffing-Van der Pol Systems by GYC Partial Region Stability Theory

A new chaos generalized synchronization strategy, using the GYC partial region stability theory, the controllers are of lower degree than that of controllers by using traditional Lyapunov asymptotical stability theorem. The simple linear homogeneous Lyapunov function of error states makes the controllers introducing less simulation error. A new Duffing-Van der Pol system and hyper-chaotic Lü system [46] are used as simulated examples.

5.1 Chaos Generalized Synchronization Strategy

Consider the following unidirectional coupled chaotic systems ( , ) vector and slave state vector respectively, f and h are nonlinear vector functions, and u=

[

u u1, , ,2 u_n

]

^T∈Rⁿ is a control input vector.

The generalized synchronization can be accomplished when t→ ∞ , the limit of the error vector e=

[

e e1, , ,2 e_n

]

^T approaches zero:

By using the partial region stability theory (see Appendix), the linear homogeneous terms of the entries of e can be used to construct a positive definite Lyapunov function and the controllers can be designed in lower degree.

5.2 Numerical Simulations

Two new Duffing-Van der Pol systems with unidirectional coupling are given:

( )

CASE I. The generalized synchronization error function is 30, 1, 2, 3, 4

i i i

e = − +x y i= (5.5) The addition of the constant 30 makes the error dynamics always happens in the first quadrant. Our goal is y_i = +x_i 30, i.e.

dynamic always exists in first quadrant as shown in Fig. 5.1. By GYC partial region asymptotical stability theory, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

1 2 3 4 which is negative definite function in the first quadrant. Four state errors versus time and time histories of states are shown in Fig. 5.2 and Fig. 5.3.

CASE II. The generalized synchronization error function is sin cos 50, 1, 2, 3, 4

2 2 we find that the error dynamics always exists in first quadrant as shown in Fig. 5.4.

By GYC partial region asymptotical stability theory, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

1 2 3 4

1 2 3 4 0

V = − − − − <e e e e (5.18) which is negative definite function in first quadrant. Four state errors versus time and time histories of states are shown in Fig. 5.5 and Fig. 5.6.

CASE III. The generalized synchronization error function is 1 3 dynamics always exists in first quadrant as shown in Fig. 5.7. By GYC partial region asymptotical stability theory, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

1 2 3 4

V = + + + (5.22) e e e e Its time derivative is

( )

which is negative definite function in first quadrant. Four state errors versus time and time histories of states are shown in Fig. 5.8 and Fig. 5.9.

CASE IV. The generalized synchronization error function is

1 150, 1, 2, 3, 4

i i i 2 i

e = − +x y z + i= (5.26)

[

1 2 3 4

]

z= z z z z T is the state vector of hyperchaotic Lü system.

The goal system for synchronization is hyperchaotic Lü system and initial states is (1, 1, 1, 1), system parametersa₁=36, b₁ =20, c₁=3, d₁ =1.3.

( )

dynamics always exists in first quadrant as shown in Fig. 5.10. By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

1 2 3 4

V = + + + (5.30) e e e e Its time derivative is

( )

which is negative definite function in first quadrant. Four state errors versus time and time histories of states are shown in Fig. 5.11 and Fig. 5.12.

 

Fig. 5.1 Phase portraits of error dynamics for Case I.

 

 

Fig. 5.4 Phase portrait of error dynamics for Case II.

 

 

Fig. 5.7 Phase portraits of error dynamics for Case III.

 

 

Fig. 5.10 Phase portrait of error dynamics for Case IV.

 

Chapter 6

Chaos Control and Anti-control of a New

Duffing-Van der Pol System by GYC Partial Region Stability Theory

Using the GYC partial region stability theory, a new chaos control and anti-control strategy is proposed. The controllers are of lower degree than that of controllers by using traditional Lyapunov asymptotical stability theorem. The simple linear homogeneous Lyapunov function of error states makes the controllers introducing less simulation error. A new Duffing-Van der Pol system and hyper-chaotic Lü system are used as simulated examples.

6.1 Chaos Control Scheme

Consider the following chaotic systems ( , )t

x f= x       (6.1) where x=

[

x x1, , ,2 x_n

]

^T∈Rⁿ is a the state vector, f:R₊×Rⁿ →Rⁿ is a vector function.

The goal system which can be either chaotic or regular, is ( , )t

=

y g y       (6.2)  where y=

[

y y1, , ,2 y_n

]

^T∈Rⁿ is a state vector, g:R₊×Rⁿ →Rⁿ is a vector function.

In order to make the chaos state vector x approaching the goal state vector y , define = −e x y as the state error. The chaos control is accomplished in the sense that [35-41]:

lim lim( ) 0

t→∞e=t→∞ x y− =     (6.3) 

In this Chapter, we will use examples in which the error dynamics is placed in the first quadrant of coordinate system and use the GYC partial region stability theory.

The Lyapunov function is a simple linear homogeneous function of error states and the controllers are simpler because they are in lower degree than that of traditional controllers.

6.2 Numerical Simulations for Chaos Control

The following chaotic system is the new Duffing-Van der Pol system of which the old origin is translated to ( , , , ) (50,50,50,50)x x x x_{1 2 3 4} = and the chaotic motion

always happens in the first quadrant of coordinate system ( , , , )x x x x . This _{1 2 3 4} translated new Duffing-Van der Pol system is presented as simulated examples where the initial states of system are x₁(0) 52, (0) 52.4, (0) 55, (0) 56= x₂ = x₃ = x₄ = and the each equation of Eq. (6.4), respectively.

( )

CASE I. Control the chaotic motion to zero.

In this case we will control the chaotic motion of the new Duffing-Van der Pol

In Fig. 6.2, we see that the error dynamics always exists in first quadrant.

By GYC partial region stability theory, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

V = + + + (6.7) e e e e Its time derivative through error dynamics (6.6) is

( ) ( ) ( ) ( ) ( )

which is negative definite function in first quadrant. The numerical results are shown in Fig.6.3. After 100 sec, the motion trajectories approach the origin.

CASE II. Control the chaotic motion to a product of sine and cosine functions.

In this case we will control the chaotic motion of the new Duffing-Van der Pol system (6.4) to a product of sine and cosine functions of time. The goal is

sin cos

In Fig. 6.4, the error dynamics always exists in first quadrant.

By GYC partial region stability theory, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

( )

which is negative definite function in first quadrant. The numerical results are shown in Fig.6.5 and Fig. 6.6. After 100 sec., the errors approach zero and the motion trajectories approach to sine and cosine functions.

CASE III. Control the chaotic motion to chaotic motion of hyper-chaotic Lü system.

In this case we will control chaotic motion of the new Duffing-Van der Pol system (6.4) to that of hyper-chaotic Lü system. The goal system is hyper-chaotic Lü system:

The error equation is 1

e = x - z . The goal is 5 ^lim_t→∞^e=0. The error dynamics

By Fig. 6.7, we know that the error dynamics always exists in first quadrant.

By GYC partial region stability theory, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

which is negative definite function in first quadrant. The numerical results are shown in Fig.6.8 and Fig. 6.9 where a₁ =36, b₁ =20, c₁=3, d₁ =1.3. After 100 sec., the errors approach zero and the chaotic trajectories of the new Duffing-Van der Pol system approach to that of the hyper-chaotic Lü system.

6.3 Numerical Simulations for Chaos Anti-control

In this section, we will control periodic motion of the new Duffing-Van der Pol system to that of hyper-chaotic Lü system. The new Duffing-Van der Pol system exhibits periodic motion when the parameters of system are

0.1, 1, 5, 0.67, 0.05

is the new Duffing-Van der Pol system of which the old origin is translated to

1 2 3 4

( , , , ) (50,50,50,50)x x x x = and the periodic motion always happens in the first

quadrant of coordinate system ( , , , )x x x x . This translated new Duffing-Van der _{1 2 3 4} Pol system is presented as simulated examples where the initial states of system are

1(0) 52, (0) 52.4, (0) 55, (0) 562 3 4 each equation of Eq. (6.18), respectively.

( )

The goal system is hyper-chaotic Lü system:

( )

The error equation is 1

e = x - z . The goal is 5 ^lim_t→∞^e=0. The error dynamics becomes

By Fig. 6.11, we know the error dynamics always exists in first quadrant.

By GYC partial region stability theory, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

( ) ( ( ) )

which is negative definite function in first quadrant. The numerical results are shown in Fig.6.12 and Fig. 6.13 where a₁ =36, b₁ =20, c₁=3, d₁ =1.3. After 200 sec., the errors approach zero and the periodic trajectories of the translated new Duffing-Van der Pol system approach to that of the hyper-chaotic Lü system.

Fig. 6.1 Chaotic phase portraits for a new Duffing-Van der Pol system in the first quadrant.

Fig. 6.2 Phase portrait of error dynamics for Case I.

 

Fig.6.3 Time histories of errors for Case I.

Fig. 6.4 Phase portrait of error dynamics for Case II.

 

Fig. 6.5 Time histories of errors for Fig. 6.6 Time histories of x for _i Case II. Case II.

Fig. 6.7 Phase portrait of error dynamics for Case III.

Fig. 6.8 Time histories of errors for Fig. 6.9 Time histories of x_i for Case Case III. III.

 

             

Fig. 6.10 Periodic phase portraits for a translated new Duffing-Van der Pol system in the first quadrant.

             

Fig. 6.11 Phase portraits of error dynamics.

                   

Fig. 6.12 Time histories of errors. Fig. 6.13 Time histories ofx . _i  

 

Chapter 7

Hybrid Projective Symplectic Synchronization of a New Duffing-Van der Pol System with Legendre function Parameters by GYC Partial Region Stability

Theory

A new type of chaotic synchronization, hybrid projective symplectic synchronization (HPSS), is obtained for a Duffing-Van der Pol system with constant parameter and a new Duffing-Van der Pol system with Legendre function parameters.

The latter is used as 〝master〞 system and the former as 〝slave〞 system. Based on the GYC partial region stability theory, the scheme can be achieved not only for projective synchronization, but also for projective anti-synchronization. Numerical simulations are provided to verify the effectiveness of the proposed scheme.

7.1 Hybrid Projective Symplectic Synchronization Scheme

There are two nonlinear chaotic dynamical systems. The 〝master 〞system controls the 〝slave〞 system partly. In symplectic synchronization, the 〝master〞

system is called partner A:

( )

where u t( ) [ ( ), ( ),= u t u t_{1 2} u t_n( )]^T∈Rⁿ is a control input vector. HPSS demands:

(

^{, ,}

)

y H x y t= =Gxy (7.4) where H x y t

(

, ,

)

consists of state vector x of partner A and state vector y of partner B, G diag g g= ( ,_{1 2},...,g_n ) ∈R^{(n n× )} is a constant scaling matrix with positive and negative entries. Our goal is to accomplish Eq. (7.4) via controller u(t). Define the error vector e:

( )

, ,

e =H x y t −y (7.5) The synchronization is achieved when

( )

By using the GYC partial region stability theory (see Appendix), the linear homogeneous terms of the entries of e can be used to construct a positive definite Lyapunov function in first quadrant and the controllers can be designed in lower degree. Hence, the HPSS can be achieved.

7.2 Chaos of a New Duffing-Van der Pol System with Legendre Function Parameters

This section introduces a new Duffing-Van der Pol system with Legendre function parameters.

( )

where a, b, c, d, f are parameters. We select the Legendre functions [48] as parameters of the system. The Legendre functions are defined by

( ) ( )

^{1 m}

(

^{1 2}

)

^{m2 m}

^{( )}

x = , the bifurcation diagram by changing constant parameter k is shown in Fig.

7.2. Its corresponding Lyapunov exponents are shown in Fig. 7.3. The phase portraits, time histories, and Poincaré maps of the systems are showed in Fig. 7.4 and Fig. 7.5.

When k=0.35, period 1 phenomena are shown in Fig. 7.4. When k=0, the chaotic behaviors are given in Fig.7.5.

7.3 Numerical Results

Since the partner A is described by Eq. (7.9). The partner B is described as

( )

and u4 are added to the partner B then becomes controlled partner B:

( )

In the HPSS, the error function is 100

e Gxy y= − + (7.16) The addition of the constant 100 makes the error dynamics always happens in the first quadrant. Our goal is y Gxy= +100, i.e.

lim lim( 100) 0

t e t Gxy y

→∞ = →∞ − + = (7.17) The error dynamics Eq. (7.8) becomes:

( )

exists in first quadrant as shown in Fig. 7.6. By GYC partial region asymptotical stability theory, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

( )

which is negative definite function in the first quadrant. The numerical result is shown in Fig. 7.7.

Fig. 7.1 Time histories of L , 1 L , and 2 L . 3

Fig. 7.2 The bifurcation diagram for new Duffing-Van der Pol system with Legendre function parameters.

Fig. 7.3 The Lyapunov exponents for new Duffing-Van der Pol system with Legendre function parameters.

Fig. 7.4 Phase portrait, Poincaré maps, and time histories for new Duffing-Van der Pol system with Legendre function parameters when k=0.35 (period 1).

Fig. 7.5 Phase portrait, Poincaré maps, and time histories for new Duffing-Van der Pol system with Legendre function parameters when k=0 (chaos).

Fig. 7.6 Phase portraits of error dynamics.

Fig. 7.7 Time histories of errors.

Chapter 8 Conclusions

 

In this thesis, the chaotic behavior in new Duffing-Van der Pol system is studied by phase portraits, time history, Poincaré maps, Lyapunov exponent, bifurcation diagrams, and parametric diagram.

Three kind of chaotic synchronization are presented. A new kind of generalized synchronization system in Chapter 3, pragmatical hybrid projective chaotic generalized synchronization (PHPCGS) of two chaotic systems with uncertain parameters, is obtained with the state variables of another hyperchaotic Mathieu-Duffing system as a constituent of the functional relation between master and slave. Based on the pragmatical asymptotical stability theorem, adaptive control law is used.

A new type for chaotic synchronization in Chapter 4, pragmatical chaotic symplectic synchronization (PCSS), is obtained with the state variables of another different order system as a constituent of the functional relation between 〝master〞

and 〝slave〞. Traditional generalized synchronizations are special cases of the symplectic synchronization. The pragmatical asymptotical stability theorem is used.

New dynamic surface control is also used for making the controllers more simple.

A new chaos generalized synchronization method, using GYC partial region stability theory, is proposed in Chapter 5. Moreover, we also study the chaos control and anti-control by using the GYC partial region stability theory in Chapter 6. By using this theory, the controllers are of lower degree than that of controllers by using traditional Lyapunov asymptotical stability theorem. The simple linear homogeneous Lyapunov function of error states makes that the controllers are simpler and introduce

less simulation error. In addition, by replacing the parameters of the system with Legendre function, the chaos synchronization can be successfully obtained in Section 7.

Appendix I GYC Pragmatical Asymptotical Theorem [29]

 

The stability for many problems in real dynamical systems is actual asymptotical stability, although may not be mathematical asymptotical stability. The mathematical asymptotical stability demands that trajectories from all initial states in the neighborhood of zero solution must approach the origin as ^{t →∞}. If there are only a small part or even one of initial states from which the trajectories or trajectory do not approach the origin as ^t→∞, the zero solution is not mathematically asymptotically stable. However, when the probability of occurrence of an event is zero, it means the event does not occur actually. If the probability of occurrence of the event that the trajectries from the initial states are that they do not approach zero when ^t→∞, is zero, the stability of zero solution is actual asymptotical stability though it is not mathematical asymptotical stability. In order to analyze the asymptotical stability of the equilibrium point of such systems, the pragmatical asymptotical stability theorem is used.

Let X and Y be two manifolds of dimensions m and n

(

^{m n<}

)

, respectively, and ϕ be a differentiable map from X to Y; then ϕ^(X) is a subset of Lebesque measure 0 of Y [47]. For an autonomous system

) For nonautonomous system,

(

1, , ,2 _n 1

)

Definition: The equilibrium point for the dynamic system is pragmatically asymptotically stable provided that with initial points on C which is a subset of

Lebesque measure 0 of D, the behaviors of the corresponding trajectories cannot be determined, while with initial points on D−C, the corresponding trajectories behave as that agree with traditional asymptotical stability[29,30].

Theorem: Let V =[x₁,x₂, x_n]^T :D→R₊ positive definite, analytic on D, where x1, ,x are all space coordinates such that the derivative of _n V through differential equation, _V, is negative semi-definite.

Let X be the m-manifold consisting of point set for which∀x ≠0, V(x)=0 and D is an n-manifold. If m+ 1<n, then the equilibrium point of the system is pragmatically asymptotically stable.

Proof: Since every point of X can be passed by a trajectory of Eq. (A.1), which is one dimensional, the collection of these trajectories, C, is a ⁽m+¹⁾-manifold [29,30]. If ^(m+1)＜ⁿ, then the collection C is a subset of Lebesque measure 0 of D.

By the above definition, the equilibrium point of the system is pragmatically asymptotically stable. □

If an initial point is ergodicly chosen in D, the probability of that the initial point falls on the collection C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighborhood of the equilibrium point. Hence, the event that the initial point is chosen from collection C does not occur actually.

Therefore, under the equal probability assumption, pragmatical asymptotical stability becomes actual asymptotical stability. When the initial point falls on D−C ,

0 ) (x <

V , the corresponding trajectories behave as if they agree with traditional asymptotical stability because by the existence and uniqueness of the solution of initial-value problem, these trajectories never meet C.

For Eq. (9) Lyapunov function V is a positive definite function of n variables, i.e.

p error state variables and n - p = m differences between unknown and estimated parameters, while _{V =e}^T_Ce is a negative semi-definite function of n variables.

Since the number of error state variables is always more than one, p>1, (m+ )1 <n is always satisfied. By pragmatical asymptotical stability theorem we have

0 lim =

∞

→ e

t (A.5) and the estimated parameters approach the uncertain parameters. The pargmatical generalized synchronizations is obtained. Therefore, the equilibrium point of the system is pragmatically asymptotically stable. Under the equal probability assumption, it is actually asymptotically stable for both error state variables and parameter variables.

Appendix II GYC Partial Region Stability Theory [42]

Consider the differential equations of disturbed motion of a nonautonomous system in the normal form

( , , , ),1 ( 1, , )

s

s n

dx X t x x s n

dt = =   (A2.1) 

where the function X is defined on the intersection of the partial region _s Ω (shown in Fig. A2.1) and that X is smooth enough to ensure the existence, uniqueness of the solution of the _s

initial value problem. When X does not contain t explicitly, the system is _s autonomous.

Obviously, x_s =0 (s=1, n) is a solution of Eq.(A2.1). We are interested to the asymptotical stability of this zero solution on partial region Ω (including the boundary) of the neighborhood of the origin which in general may consist of several subregions (Fig. A2.1).

, x

s

s2<ε

∑

(s=1,… , n) (A2.4)

is satisfied for the solutions of Eq. (A2.1) on Ω , then the disturbed motion

0 ( 1, )

xs = s= n is stable on the partial region Ω . Definition 2:

If the undisturbed motion is stable on the partial region Ω , and there exists a

' 0

is satisfied for the solutions of Eq.(A2.1) on Ω , then the undisturbed motion

0 ( 1, )

xs = s= n is asymptotically stable on the partial region Ω .

xs = s= n is asymptotically stable on the partial region Ω .

在文檔中 新Duffing-Van der Pol 系統的渾沌現象，實用渾沌廣義同步，辛同步，應用GYC部分區域穩定理論的渾沌同步及控制 (頁 34-0)